论文标题
通用阿贝尔表面的同时学，并应用于算术统计
Cohomology of the Universal Abelian Surface with Applications to Arithmetic Statistics
论文作者
论文摘要
原则上极化的Abelian表面的Moduli堆栈$ \ MATHCAL A_2 $配备了通用Abelian Surface $π：\ Mathcal x_2 \ to \ Mathcal A_2 $。 $π$的光纤在对应于Abelian Surface $ a in $ \ Mathcal a_2 $的点上是$ a $本身。我们确定$ \ Mathcal X_2 $作为Galois表示的$ \ Ell $ - adic共同体。同样，我们考虑捆绑$ \ MATHCAL x_2^n \ to \ MATHCAL A_2 $和$ \ MATHCAL X_2^{\ propatatorName {sym}（sym}（n）} \ to \ Mathcal a_2 $ to All $ n \ geq 1 $，在所有$ n \ geq 1 $上，$ n \ egq 1 $，$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ $ \ operatatorName {sym}^n a $。我们描述了如何计算$ \ Mathcal x_2^n $和$ \ Mathcal x_2^{\ propatatorName {sym}（n）} $的$ \ ell $ - adic共同体，并在所有$ n $中以$ n = $ n = 2 $进行低度。这些结果产生了有关阿伯利亚表面上算术统计的新信息，包括对$ \ MATHBF F_Q $ - 点的较高矩的预期值和方差的精确计算以及渐近学。
The moduli stack $\mathcal A_2$ of principally polarized abelian surfaces comes equipped with the universal abelian surface $π: \mathcal X_2 \to \mathcal A_2$. The fiber of $π$ over a point corresponding to an abelian surface $A$ in $\mathcal A_2$ is $A$ itself. We determine the $\ell$-adic cohomology of $\mathcal X_2$ as a Galois representation. Similarly, we consider the bundles $\mathcal X_2^n \to \mathcal A_2$ and $\mathcal X_2^{\operatorname{Sym}(n)} \to \mathcal A_2$ for all $n \geq 1$, where the fiber over a point corresponding to an abelian surface $A$ is $A^n$ and $\operatorname{Sym}^n A$ respectively. We describe how to compute the $\ell$-adic cohomology of $\mathcal X_2^n$ and $\mathcal X_2^{\operatorname{Sym}(n)}$ and explicitly calculate it in low degrees for all $n$ and in all degrees for $n = 2$. These results yield new information regarding the arithmetic statistics on abelian surfaces, including an exact calculation of the expected value and variance as well as asymptotics for higher moments of the number of $\mathbf F_q$-points.
